When we first made this observation we were unaware of the seminal work of Carlip[1]}, which had proposed that the horizons of black holes were described by such a CFT. That work was motivated by the invariance of the near horizon geometry under a Virasoro subalgebra of the algebra of infinitesimal diffeomorphisms. Using the symplectic structure of the Einstein-Hilbert Lagrangian, the Poisson algebra of these generators has a central charge and a classical value of the \(L_0\)  generator, which reproduce the Bekenstein-Hawking entropy for all Minkowski black holes, if one uses the Cardy formula. In fact, this derivation works for the cosmological horizon of dS space, or any causal diamond smaller than the radius of curvature in maximally symmetric space-times. In[2]} we conjectured that this cutoff \(1 + 1\)  CFT is a general signature of the entanglement spectra of causal diamondsThe analysis of [1]} is ultralocal on the co-dimension two holographic screen of a diamond.
For large causal diamonds or black holes in AdS space, the spectrum of quasi-normal modes includes sound modes, which indicate that there are more global contributions to the entropy. The black hole entropy formula for such diamonds, validated by the AdS/CFT correspondence, shows that the entropy is actually dominated by the entropy of generalized sound modes. In tensor network language, the horizon looks like a lattice of \(1 + 1\)  dimensional CFTs of large central charge, with short range couplings between them..
This picture suggests that the Hamiltonian describing the equilibrium states of dS space is a cutoff \(1 + 1\)  dimensional CFT, identical to that describing a Minkowski black hole of the same radius. The Hamiltonian should also incorporate the property of fast scrambling[4]} of information on the dS horizon. This is achieved if the geometry of the holographic screen is related to the target space of the \(1 + 1\)  CFT and the dynamics is invariant under area preserving diffeomorphisms of the screen. The finite total entropy then implies that the algebra of measurable functions on the screen must be finite dimensional.
